On the mathematical foundations of Syntactic Structures Geoffrey K. Pullum School of Philosophy, Psychology and Language Sciences University of Edinburgh 5 April 2011 Abstract Chomsky’s highly influential Syntactic Structures (SS) has been much praised its originality, explicitness, and relevance for subsequent cognitive science. Such claims are greatly overstated. SS contains no proof that English is beyond the power of finite state description (it is not clear that Chomsky ever gave a sound mathematical argument for that claim). The approach advocated by SS springs directly out of the work of the mathematical logician Emil Post on formalizing proof, but few linguists are aware of this, because Post’s papers are not cited. Chomsky’s extensions to Post’s systems are not clearly defined, and the arguments for their necessity are weak. Linguists have also overlooked Post’s proofs of the first two theorems about effects of rule format restrictions on generative capacity, published more than ten years before SS was published. Keywords Generative grammar · transformations · Emil Post · formalization · proof theory · mathematical logic 1 Introduction In 1957, when Martin Joos first published his classic anthology of American structuralism, Readings in Linguistics I (1957), it was offered as a contribution to a flourishing research program. The same could be said about J. R. Firth’s collected Papers in Linguistics 1934- 1951 (Firth 1957) and the Philological Society’s definitive anthology of London-school lin- guistics, Studies in Linguistic Analysis (Philological Society 1957), or about B. F. Skinner’s long-awaited book about verbal behavior based on his William James Lectures from ten years earlier (Skinner 1957). In retrospect, however, these works look like valedictions: concluding summaries of paradigms that had reached their use-by dates. In December 1957, Language published an extraordinarily laudatory review (Lees 1957) of a short monograph that had been published on February 14 that year in a new series from a small publisher in Holland. The book was Syntactic Structures (Chomsky 1957, henceforth SS). The author was at the time an unknown 28-year-old who taught language classes at MIT. The reviewer, Robert B. Lees, hailed SS as a revolutionary scientific breakthrough, and from 1958 on, linguists began to pay it a great deal of attention. Lees’s claims about revolution are controversial. Newmeyer (1986) argues that SS did indeed spark a scientific revolution, and others disagree. I take no position here on that question (a sociological one, as Newmeyer construes it). I will argue, however, that many exaggerated claims have been made about SS, some of them straightforwardly false. For example, the claim that in SS Chomsky gave a “proof” that “demonstrated the inadequacy This paper has been published online by the Journal of Logic, Language and Information (at the http://www.springerlink.com website; DOI 10.1007/s10849-011-9139-8), and will subsequently appear in the print edition. This is my own text, differeing in pagination from the version published by JoLLI. To contact the author: E-mail: [email protected] or phone +44-131-650-3603. 2 of finite state grammars” (Lyons 1970: 54) is not true: Chomsky did not even attempt such a proof in SS (see section 2 below). Recent claims about the content and effects of SS have been getting more extreme rather than less. Josie Glausiusz says that Chomsky “in the 1950s proposed that all humans are equipped with a universal linguistic grammar, a set of instinctive rules that underlie all lan- guages” (Discover magazine, May 2007). But Chomsky made no proposals about innate universal grammar in the 1950s. David Lightfoot calls SS “the snowball which began the avalanche of the modern ‘cognitive revolution’ [that] originated in the seventeenth century and now construes modern linguistics as part of psychology and human biology” (Light- foot 2002: v). But SS contains not even a nod in the direction of the study of cognition or 17th-century thought. When the reputation of a book gives rise to this kind of exaggeration and misattribution, re-evaluation is called for. The task is a large one, and I do not aim to cover all aspects of SS in this paper.1 My focus is on selected aspects of the mathematical and formal language- theoretic underpinnings of SS. 2 The purported proof that English is not finite-state It is widely believed that SS gives a proof that the stringset of all syntactically well-formed sequences of English words is not a finite-state language (FSL). It does not. SS never at- tempts a rigorous argument; it just informally adumbrates one, and then asserts that “it is impossible, not just difficult, to construct a device of the [finite automaton] type . which will produce all and only the grammatical sentences of English” (SS: 23). A weak generative capacity proof is required to show that English is not an FSL, and SS did not give one. Of course, SS was deliberately trying to keep things elementary—it originated as the notes for a series of lectures for undergraduate scientists and engineers at MIT, and was supposed to offer an informal digest of a technical paper, Chomsky (1956b), and a much larger unpublished typescript, Chomsky (1956a, published much later with revisions and omissions as Chomsky 1975). But in fact the earlier works do not contain a demonstration of the non-FSL character of English either. SS (p. 22) cites Chomsky 1956b, claiming it contains the “rigorous proof” to which SS alludes, but this is not so. In the form originally given in Chomsky (1956b) it depended on a cumbersomely defined relation of “(i, j)-dependency” holding between a string S of length n, two integers i and j such that 1 ≤ i < j ≤ n, and a language L over a vocab- ulary A, and the attempted argument was not valid. The definitions are changed in the 1965 reprint version of the paper (a footnote credits E. Assmuss for pointing out a de- fect in the original formulation): this revised version relies on a cumbersomely defined ternary relation of ‘m-dependency’ between a sentence S, an integer m, and a stringset L, where S = x1 a1 x2 a2 ... xm am zb1 y1 b2 y2 ... bm ym, and there is a unique mapping of the set {1,...,m} to itself (a “permutation”) meeting the following condition (I quote from p. 108): there are {c1,...,c2m}∈ A such that for each subsequence (i1,...,ip) of (1,...m), S1 is not a sentence of L and S2 is a sentence of L, where (10) S1 is formed by substituting ci j for ai j in S, for each j ≤ p; S2 is formed by substituting cm+α(i j) for bα(i j) in S1, for each j ≤ p. The idea is that if in the string S the symbol ai is replaced by the symbol ci, restoring grammaticality in L necessitates replacing bα(i) by cm+α(i). From there, the crucially relevant mathematical step is to claim that an FSL can only ex- hibit m-dependencies up to some finite upper bound on m (Chomsky says an m-dependency 1 For example, I will not discuss here the critique of statistical approaches to grammaticality (SS: 15–17), though it was so influential that the now burgeoning work on stochastic approaches to grammar virtually disappeared from the scene for thirty years. The claim that probabilistic models can never distinguish gram- matical but unlikely nonsense sequences from ungrammatical sequences (SS: 16) was not true, but it was not properly answered until Pereira (2000) showed what a huge difference it makes when a crude statistical model that assigns zero probability to anything not yet attested is replaced by one that uses Good–Turing ‘smoothing’. I also omit discussion of the philosophical and historical background; for that, see Tomalin (2006), Scholz & Pullum (2007), and Seuren (2009). 3 needs at least 2m states; Svenonius 1957 says this is untrue, and m states will suffice). The empirical claim is that English has no such upper bound, and is therefore not an FSL. But Chomsky does not complete the argument by connecting these abstractions to En- glish data; he merely points to some sentence templates (“If S1, then S2”; “Either S3, or S4”; “The man who said that S5, is arriving today” [comma in original]), and asserts that through them “we arrive at subparts of English with . mirror image properties” and thus “we can prove the literal inapplicability of this model” (Chomsky 1956b, 1965 reprinting, p. 109). Daly 1974 spends many pages attempting to work out how a sound argument for Chom- sky’s conclusion might be based on the data that he cites. Chomsky seems to think that pairs like if, then and either, or give rise to m-dependencies. Daly could not see how this could be true. Nor can I. The words in these pairs can occur in sentences without the other member of the pair. (The same is true of other pairs such as neither, nor and both, and.) It is not clear that there is any pair of lexical items σ and τ in English such that if ϕσψ is grammatical then ψ = ψ1 τ ψ2 with |ψ1| > 0. In addition, remarks like “we can find various kinds of non-finite state models within English” (SS: 22–23) and the similar remark that “we arrive at subparts of English with . mirror image properties” (Chomsky 1956b, 1965 reprinting, p. 109) suggest a failure to appreciate that FSLs can have infinite non-finite-state subsets. Only if such a subset can be extracted by some regularity-preserving operation like homomorphism or intersection with a regular set does it entail anything about the language as a whole.